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On tempered and substantial fractional
calculus
YangQuan Chen
Collaborators: Jianxiong Cao, Changpin Li
School of Engineering,
University of California, Merced
Department of Mathematics, Shanghai University
E: yangquan. chen@ucmerced.edu
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Outline
• Preliminaries of fractional calculus
• Tempered and substantial fractional operators
• Numerical solution of a tempered diffusion
equation
• Conclusion
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Preliminaries
• Calculus=integration+differentiation
• Fractional Calculus= fractional integration+
fractional differentiation
• Fractional integral:
Mainly one: Riemann-Liouville integral
• Fractional derivatives
More than 6: not mutually equivalent
RL and Caputo derivatives are mostly used
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Preliminaries
• Left Riemann-Liouville integral
• Right Riemann-Liouville Integral
• Left Riemann-Liouville fractional derivative
• Right Riemann-liouville Fractional derivative
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• Left Caputo fractional derivative
• Right Caputo fractional derivative
• Riesz fractional derivative
• Riesz-Caputo fractional derivative
where
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Tempered and substantial operator
Let
be piecewise continuous on
integrable on any subinterval of
,and
,
1) The left Riemann-Liouville tempered fractional
integral is
2) The right Riemann-Liouville tempered fractional
integral
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(Phys. Rev. E 76, 041105, 2007 )
Suppose that
be
times continuously differentiable on
,
and times derivatives be integrable on any subinterval
of
, then the left and right Riemann-Liouville tempered
derivative are defined
and
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Remark I: From the above definitions, we can see if
then they reduce to left and right Riemann-Liouville
Fractional derivatives.
Remark II: The variants of the left and right Riemann-Liouville
tempered fractional derivatives (Boris Baeumer, JCAM, 2010)
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(Phys. Rev. Lett. 96, 230601, 2006)
Substantial fractional operators
Let
be piecewise continuous on
, and integrable on
any subinterval of
, then the substantial fractional
integral is defined by
And the substantial derivative is defined as
where
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Theorem:
If the parameter
is a positive constant, the tempered
and substantial operators are equivalent.
See our paper for the details of proof.
Recently, tempered fractional calculus is introduced in
Mark M. Meerschaert et al. JCP(2014).
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Numerical computation of a tempered diffusion
equation
We consider the following tempered diffusion
equation
where
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The exact solution of above problem is
We use finite difference method and shifted Grunwald
method to numerically solve it, we choose different value
Of parameter
, and compare the results with its exact
solution
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 From the figures, we can see that the
numerical solution are in good accordance
with exact solutions.
 It is easily shown that the peak of the
solutions of tempered diffusion equation
becomes more and more smooth as
exponential factor λ increases.
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Conclusion
• Tempered and substantial fractional calculus are
the generalizations of fractional calculus.
• The two fractional operators are equivalent,
although they are introdued from different
physical backgrounds.
• Tempered fractional operators are the best tool for
truncated exponential power law description
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Thanks for your attention